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Saturday, February 17, 2007

I'm thinking....I might have to have a talk with the principal about inflammatory language.

After all, he's had 3 talks with me on the subject. He has, in fact, "urged" me to be "constructive."

I'm thinking it's time to make the point that things like refusing to provide the answer key to a parent who is actually planning to use the eleven-dollar state test prep booklet the school has required her to buy are inflammatory.

They breed ill will.

They lead directly to bond proposals getting voted down and middle school models getting shot down.

I'm thinking that pointing this out in the least inflammatory, most congenial tone of voice I can possibly muster might be a very good idea.

I love the way Singapore math approaches word problems. I use it for all my students including tutoring college students.

When I teach word problems to my math booster classes, I teach them the Singapore math model of all parts add to a whole. So you draw a rectangle to represent the whole. You draw a rectangle of exactly the same size below the first one but break it into the number of parts listed in the problem. If you are given all the parts in the problem, you add them to get the whole. If you are given the whole, then you will either be given all but one part or you will be given the relationship between the parts.

For example:

Your mom bought 6 apples and 8 pears. How many pieces of fruit did she buy? This gives 2 parts that you add to get the total.

You mom bought 14 apples and pears. She bought 6 apples. How many pears did she buy? This is the same problem, but you are given the total and one part and you have to subtract to get the second part.

Now, in Singapore math, the complexity increases when you are given a total and the relationship between parts.

Your mom buys 14 apples and pears. She buy 2 more apples than pears. How many apples did she buy. You are give a total and the relationship between the two parts. The beauty of the Singapore math system is that the rectangles that you draw, one on top of another, show that you take 2 away from 14 and divide 12 by 2 to get the number of pears and add two to get the number of apples.

'Classic' problems in algebra like the ones you describe (coin, mixture, rate) are done exactly the same way except that you must also practice dimensional analysis. The numbers in the problem are not necessarily directly additive.

For example, a 'classic' rate problem is this: two trains depart from different cities heading towards each other. One is going 55mph and one is going 65mph. If the cities are 330 miles apart, how long will it take them to pass each other?

This is still a word problem with two parts adding to one total. But the total is in miles and the parts are given as a rate. Rates are not additive, so the rates must be converted into miles and the miles from the two different trains have to add to 330. The relationship between the two parts is given by time since when they pass each other, each has been traveling the same amount of time. So 55t+65t=330. Solve for t.

What this shows is that by the time Singapore students get to algebra, they already have a method to solve all the word problems. All they have to do is substitute a letter for one of the boxes.

Cool!

Thanks!

Of course this motivates me to drag Christopher by the hair back to the Challenging Word Problems book.

I've gone through C's 343-page state test prep book - that's 343 pages sans answers - which was sent home with the kids with instructions to "use the book."

We're using the book.

I figure C should do 3 lessons a day over break (test starts March 12). In theory the answer key will arrive on Wednesday, so that means I need to work the problems in 12 lessons.

Which I am doing.

Ed and I had a spat about this yesterday. I had planned to refuse to purchase the test prep book unless the school gave me the answer key, too.

Ed managed not to notice that I'd been plotting this move for weeks, and sent in the $11 with Christopher when asked.

So now we have the book, and the principal can just say no when I ask for the answers. (He already had said no when I brought it up a couple of weeks ago, shortly before he also said no to moving Christopher into the Phase 3 class for the remainder of the year and then back into Phase 4 come fall. "No" is the word!)

Yesterday Ed told me he "completely disagreed" with my plan to refuse to buy a test prep book with no answers; that would mean the principal would simply not sell me the book; etc.

I said I have books; I have at least $200-worth of books, all of them with answers in the back.

He said, Those books aren't targeted to the material being tested.

etc.

The fact is, it would make my life much easier to have a test prep book specifically constructed to provide review and practice of New York state math standards for grade 7.

It does not make my life easier to have such a book without the answers.

Ed's attitude was that I could just work the problems myself.

So.

Today he's sorry.

When he left 20 minutes ago to go to Stew Leonard's and Costco I was still working on state test prep problems.

I pointed this out (I'm going to be pointing it out for days) and Ed said, "Are you doing all the problems in the book?"

No.

I'm doing the first 12 lessons in the book.

This is how long it takes to do the first 12 lessons in the book.

etc.

Who's sorry now?!

His parting shot was an offer to do some of the problems himself.

I ought to take him up on that.

The reason I ought to take him up on that is that the first lessons in the book, the ones I'm doing, are on probability and he has no idea how to do problems on probability. I bet he doesn't even remember how to construct a proper tree diagram!

Which brings me to the reason I'm writing this post.

I can't do this problem:

Raymond has a deck of 5 cards, numbered 3-7. How many different ways can the cards be lined up?

I hope the question of using a curriculum that’s heavy in word problems will get serious attention from researchers.

Singapore Math uses what people call a "problem-solving approach" from the get-go.

I think it's probably more accurate to call it an "applications" approach. Every lesson and every problem set requires students to apply the concept or procedure to something concrete (i.e. "real world").

Saxon is much more abstract. (I’ve wondered whether girls might be happier with Singapore & boys with Saxon, but I have no idea. For what it's worth I remember being struck by Charles Murray's observation that women are underrepresented in disciplines that are highly abstract, i.e. physics, math, music.)

Even when Saxon teaches the classic word problems (mixture problems, coin problems, etc.) he teaches them as representatives of a class. He takes an abstract approach to word problems.

This makes Saxon very different in feel at least from an “everyday math” approach (no caps).

teaching the classic word problems

I’ve had one experience of Saxon’s approach to word problems “working” for me.

I was taught almost none of the classic word problems in my own high school algebra courses. No number problems, no coin problems, no what time do two trains meet problems, no mixture problems, possibly not even any work problems. (And I took two years of algebra. I must have had the original dumbed-down curriculum.)

I got so I could do coin problems in my sleep....and, somewhere along the line, I realized I could do simple mixture problems by analogy to coin problems.

That was pretty interesting. I remember reading Saxon saying that there's a reason why the classic word problems are classics; my experience of making a leap from coin problems to mixture problems seems like evidence that he's right.

John Saxon on the classic word problems

Books will de-emphasize the teaching of radical expressions, conic sections, paper-and-pencil solutions of trigonometric equations, and the solutions of the old-fashioned fundamental word problems that have been used historically to teach the concepts and skills necessary to solve all problems.

I've been told by more than one parent that Trailblazers looks impressive because they see their kids doing word problems, which they always found impossible.

But I wonder whether the reason word problems were so hard for so many had to do with inadequate instruction, premature demands for generalization, etc., not with some intrinsic challenge posed by word problems per se.

In theory, word problems ought to make math easier. The lack of word problems in the Phase 4 class has been a constant issue precisely because the lack of word problems makes the course harder. The kids get very few concrete illustrations of how the math works.

Here's a blogger talking about word problems making math easier:

It sure seems like a lot of folks absolutely hate ‘word problems‘ - you know, math problems that use words in addition to all of those pesky, confusing numbers. I dunno…word problems always seemed to be much easier for me than regular “numbers-only” math…having real-world examples to work with and provide imagery helped me quite a bit.

That's the way Singapore Math uses word problems - to provide real-world examples and imagery that make the concept more manageable.

I think we may all be awed by a curriculum that seems hard given what we had when we were kids.

In reality, the part of Singapore Math that seems "hard" may be the easy part.

Office Depot here is selling tip cheat cards by the cashier -- for those who can't calculate a 15% tip. A locally-owned video store is having a clearance sale, 50% off. They have big signs next to the clearance aisles that tell you for each video price what 50% off would be.

Ever since taking earth science courses in college I have been fascinated by the fusion processes in distant stars that produce different elements. This gold story explains why gold is so rare, as rare as a good education.

++++++On a housekeeping note: The Recent Comments feature is frozen in time. I am addicted to my daily dose of comments, and am suffering from withdrawal symptoms. I find it relaxing to come home from an exhausting day of teaching and read the latest posts and comments. Maybe it's time to add the Haloscan feature which can display a longer list of comments. Techies?

Friday, February 16, 2007

A practical guide + a few minutes = success! There are more than 33.6 million kids in grades K 8 in public schools. So think how many parents are struggling to help their kids with the dreaded math homework.

[snip]

Features include model problems to help parents zero in on the right topic, many sample problems with step by step solutions, and a glossary of relevant terms.

Ms. Urban used to send home Xeroxed copies of the answer key, but now that Ms. Urban has retired her policy seems to have retired with her. Parents have requested answers be sent home; department’s answer has been ‘no.’

As far as the test prep books are concerned, the problem seems to be copyright law. Legally speaking the school should not be Xeroxing copies of the Teacher Guide to pass out to parents.

That problem is easily solved, I would think. In future years the school can find out how many parents wish to purchase Teacher Guides and order accordingly.

Or perhaps the math department can look at establishing an informal policy of having all students check and correct their work using the answer key in the Teacher Guide. In that case the school would ask everyone to purchase both books as a matter of course.

In any event, Irvington parents can order their own copies directly from Triumph Learning if they like. Office is closed on Monday, open again on Tuesday.

Catherine J.

direct instruction in Irvington

This is funny. After I talked to the rep at Triumph Learning, I read the "About Triumph Learning" page and found this:

State-specific Coach books are systematically developed. First, the state’s curriculum standards and previous tests are thoroughly reviewed by Triumph editors for content. Then the tests are analyzed for question style and format, typography, and narrative characteristics. Testing coordinators and specialists from the individual states are engaged to review and comment on the initial samples prepared by writers; the samples are then revised based upon this input. As writing proceeds, educators at the state department level and from the larger districts within the state are asked to review and critique the materials.

Coach books are carefully structured to match the standards and skills in each state’s curriculum. These standards and skills are broken down into their component parts and each is explicitly taught in the student books through example, instruction, and practice. Practice tests are included as well and, for the most part, the books are self-instructional. [ed.: except for that little problem about being able to check & correct your work, of course]

This is hilarious.

We've got the science department telling parents that only the top 10% of students in the country can handle the 8th grade Regents Earth Science* course because it's "taught conceptually."

If the state told the school they had to get everyone passing the Earth Science Regents test at the end of 8th grade there'd probably be a whole lot less talk about "explorations" and "investigations" and a whole lot more talk about breaking concepts down into their component parts and teaching them explicitly through example, instruction, and practice.

More state standards, please!

___________________*Regents courses are required for high school graduation. The course being taught in 8th grade is the same course everyone takes later on, normally in 10th grade. One parent asked how students below the 90th percentile were "miraculously" going to be able to handle the course in high school and the answer was that they would be "more mature." As a consequence of this exchange I have now discovered the presence of raging developmentalism in my school district.

Under the auspices of the National Research Council, this committee’s charge was to evaluate the quality of the evaluations of the 13 mathematics curriculum materials supported by the National Science Foundation (NSF) (an estimated $93 million) and 6 of the commercially generated mathematics curriculum materials (listing in Chapter 2).

The committee was charged to determine whether the currently available data are sufficient for evaluating the effectiveness of these materials and, if these data are not sufficiently robust, the committee was asked to develop recommendations about the design of a subsequent project that could result in the generation of more reliable and valid data for evaluating these materials.page 1

The following 13 mathematics curricula programs1 (The K-12 Mathematics Curriculum Center, 2002) were supported by the NSF, and the evaluations of these materials were reviewed by our committee:

These 19 curricular projects essentially have been experiments. We owe them a careful reading on their effectiveness. Demands for evaluation may be cast as a sign of failure, but we would rather stress that this examination is a sign of the success of these programs to engage a country in a scholarly debate on the question of curricular effectiveness and the essential underlying question, What is most important for our youth to learn in their studies in mathematics? To summarily blame national decline on a set of curricula whose use has a limited market share lacks credibility. At the same time, to find out if a major investment in an approach is successful and worthwhile is a prime example of responsible policy. In experimentation, success and worthiness are two different measures of experimental value. An experiment can fail and yet be worthy. The experiments that probably should not be run are those in which it is either impossible to determine if the experiment has failed or it is ensured at the start, by design, that the experiment will succeed. The contribution of the committee is intended to help us ascertain these distinctive outcomes.

The charge to the committee was “to assess the quality of studies about the effectiveness of 13 sets of mathematics curriculum materials developed through NSF support and six sets of commercially generated curriculum materials.”

[snip]

In response to our charge, the committee finds that:

The corpus of evaluation studies as a whole across the 19 programs studied does not permit one to determine the effectiveness of individual programs with high degree of certainty, due to the restricted number of studies for any particular curriculum, limitations in the array of methods used, and the uneven quality of the studies.page 188

The National Academies Press (NAP) was created by the National Academies to publish the reports issued by the National Academy of Sciences, the National Academy of Engineering, the Institute of Medicine, and the National Research Council, all operating under a charter granted by the Congress of the United States. The NAP publishes more than 200 books a year on a wide range of topics in science, engineering, and health, capturing the most authoritative views on important issues in science and health policy. The institutions represented by the NAP are unique in that they attract the nation's leading experts in every field to serve on their award-winning panels and committees. This is the right place for definitive information on everything from space science to animal nutrition.

My Human Development class at ed school uses the book, "Educational Psychology" by Ormrod. It says that students with learning disabilities or ADHD "may lack some of the basic concepts and skills that their nondisabled peers have already acquired and so need individualized instruction and practice to fill in the gaps."

So far so good, but fasten your seat belts:

"Many theorists suggest that one or another form of direct instruction, whereby students are specifically taught the things they need to learn, can be especially effective."

What a novel concept! Teach students the things they need to learn. What does this say about how the students are taught who aren't learning disabled or ADHD?

Whoohoo! I just learned my school district has decided NOT to pilot TERC as they had planned for this spring. Don’t know any more details, but I will say that some choice emails I sent to some parents have been getting a bit of attention. I just emailed the Ohio article out.

Word around here is that TRAILBLAZERS will be gone in 3 years' time. It was first implemented in school year 2004-2005; two and a half years later it's on life support, large numbers of K-5 parents are activated and angry, a school bond has been defeated.

School districts need to ask themselves whether they need this kind of headache.

Yes, administrations and school boards have the power to impose a curriculum by fiat, as happened here. Three hundred parents formally protested the adoption of TRAILBLAZERS; the administration adopted it anyway.

But there is a cost to rolling over parents and taxpayers. Power isn't free.

how politics work

I have no idea how politics work; I wish I did.

However, in the past two years of writing and reading ktm I think I've discovered the arguments that work - or have the best shot at working:

who's the expert? if mathematicians reject these programs, the school must reject these programs, too; at a minimum the school must not force these programs on parents who object. Teachers have expertise in teaching; mathematicians have expertise in math. When teachers and mathematicians disagree, mathematicians' word goes.

compare and contrast: schools must adopt curricula that will bring our children up to the level of their peers in Europe and Asia, which means curricula that allow our children to study and master algebra in the 8th grade. Can a TERC, EM, or TRAILBLAZERS student pass the Singapore placement test for 6th grade?

choose your battles: Does the school really want the kind of headaches these programs bring with? Do school administrators and board members want angry emails from parents? Do they want news stories about parents taking their kids to KUMON? Do they want citizens wondering out loud why they have to pay taxes for TERC-EM-TRAILBLAZERS and for tutors to remediate TERC-EM-TRAILBLAZERS? Does any of this sound fun?

With math, I don't talk "research shows." Research doesn't show; the research on math instruction is nowhere near the broad scientific consensus that exists on early reading instruction.

So I don't talk "research shows," and I'm unwilling to listen to an administrator talking "research shows" to me. There is no research.

I talk mathematicians, I talk international comparisons, and I talk horse sense. Horse sense meaning I don't care what the NCTM has to say; it makes no sense to start a war with parents over a math curriculum.

Speaking of wars with parents, because I strongly favor parent choice I don't support removing TRAILBLAZERS by fiat, either. We have parents who feel TRAILBLAZERS is working well for their children; one of my closest friends, a mom who knows math, is a fan of the program.

If TRAILBLAZERS is going to go it should be phased out the same way it was phased in - or it should remain as an option if we have a core of parents who want it for their younger children.

Parent choice.

Parent choice means that parents who have fared well with TRAILBLAZERS should have TRAILBLAZERS.

And parent choice means that those of us who want a curriculum endorsed by mathematicians should have a curriculum endorsed by mathematicians.

Supposedly the goal of education is higher order thinking which can be defined as:

A complex level of thinking that entails analyzing and classifying or organizing perceived qualities or relationships, meaningfully combining concepts and principles verbally or in the production of art works or performances, and then synthesizing ideas into supportable, encompassing thoughts or generalizations that hold true for many situations

It's commonly thought that these higher order thinking skills can be taught directly apart from the relevant domain knowledge (i.e., a narrow portion of knowledge that deals with the specific topic of interest). The thought is that you don't need to learn (i.e., memorize) all those messy facts because you can use your fancy higher order thinking skills to figure out whatever you need to know. Thus, instructional time is concentrated on higher order thinking skills and the learning of facts is downplayed.

Let's put that theory to the test.

No doubt if you enjoy reading (or at least take the time to read) an obscure education blog you went to college, are highly educated, and are smarter than the average bear. In other words, you have higher order thinking skills in spades. Let's test how well you can use them.

Consider the following:

You have two identical glasses, both filled to exactly the same level. One contains red dye, the other water. You take exactly one spoonful of red dye and put it in the water glass. Then you take one spoonful of the mixture from the water glass and return it to the red dye glass.

Question: Is there more red dye in the water glass than water in the red dye glass? Or is there more water in the red dye glass than red dye in the water glass? In other words, the percentage of foreign matter in each glass has changed.Has the percentage changed more in one of the glasses, or is the percentage change the same for both glasses?

Use your superior higher order thinking skills and intuit an answer. First try to do it without resorting to outside sources. Then try answering it using whatever reference source is handy, such as google.

NB: This only works if you don't know the scientific principle involved. If you happen to know the right scientific principle, you're relying on your domain knowledge to answer the question, not your higher order thinking skills. Also, no fair if you know the source of this problem.

I'll let you stew on it for a while. I'll update later today.

Partial Update:Hint--Instead of water and red dye, think of red balls and white balls. Assume that each glass starts out with 100 balls of a single color. Now remove a number of red balls from the red-ball glass and put them in the white-ball glass. Then return the same number of balls from the glass with the “mixture” and put them in the red-ball glass. Do this with different numbers of red and white balls.

I've come to realize that probably one reason I struggled with algebra, geometry et.al., was that it seemed to me that these were basically reactionary academic disciplines, useful for designing weaponry or potentially repressive computer technology, but not with any obvious humanistic or social positive uses.

If I'm wrong about this, I'd appreciate it if people could show me how this discipline can have progressive uses.

I also feel this could be useful in developing better ways of teaching higher mathematics if such uses could be found.

...[The Commission's] plan is to "require all teachers to produce student learning gains and receive positive principal or teacher peer review evaluations to meet the new definition of a Highly Qualified and Effective Teacher (HQET)."

what he said

Observe here the basic flaw in the Commission's approach: start with a sound instinct (gauging teachers' effectiveness by their impact on pupil achievement). Ignore how little is known for sure about reliable ways of doing this, especially at scale. Then pretend that the U.S. Department of Education is capable of (and has the clout to) shape and micromanage such complicated processes as vetting teacher performance from Washington. Along the way, neglect to undo the mistakes of NCLB, so that instructors must still meet the current law's paperwork-laden, credential-heavy "highly qualified teachers" requirements (which mostly serve to keep talented people out of the classroom) even if they do prove effective at boosting student achievement. The likely result? If past is prologue, Congress and the Education Department will muck up this entire enterprise, setting back a promising idea (evaluating teachers based on their impact on student learning) for a generation.

I used inverse operations. I added 5 to both sides of the equation because addition and subtraction are inverse operations (y/7 - 5 + 5 = 4 + 5, so y/7 = 9). Then I multiplied both sides by 7 because multiplication and division are inverse operations (y/7 x 7 = 9 x 7, so y = 63).

For me, this is perfect.

It's actually what I was reaching for....and just couldn't get hold of. I kept getting jumbled up trying to remember "additive identity" (or whatever it is)....I knew there was something simpler (is it simpler? I don't know!)....

Anyway, "I used inverse operations" was the description I was trying to find.

It's a significant step up in level of abstraction from "I added, then I subtracted," but it still sounds like something a 7th grader could say (in a perfect world) & doesn't require the student to know that math facts are theorems.

The other great thing about this answer is that it's the central point I've been trying to make with C. lately, viz.: solving an algebra equation means "undoing" what's been done to the number. (I recall Steve not being keen on this definition, and I certainly can't argue the case. I'm attempting to teach it to Christopher because it worked so well for me when I read Carolyn's post. Ed says he was taught that algebra was "undoing" when he was in high school, too.)

Christopher solved the equation easily enough. But then, for "explain how you found the value of y," he wrote "I guessed and checked." Which he manifestly had not done, seeing as how he'd scribbled some calculations on the paper that clearly revealed the fact that he had added 5 to 4, then multiplied 9 by 7.

I wanted to bean him.

Apparently adolescence gets worse before it gets better. Lately Christopher has taken to imitating me and hurling challenges such as "Do you ever think about anything except math?" my way on a minutely basis.

That's "minutely" pronounced min - et - lee, by analogy to "hourly" or "daily." I am experiencing challenges to my authority on a minute-by-minute basis, as opposed to an hourly or a daily basis.

On four of the 11 questions, at least half of the students got them wrong. Students did not receive a grade.

Remediation becomes necessary:

Each of Dublin’s four middle schools is implementing an emergency plan to tutor students this school year. Some will take up to 20 minutes of class time to review basic computation. Others will give algebra students extra homework in division.

Parents hire math tutors:

Some Dublin parents said they have spent hundreds on tutoring services.

Local Kumon centers see enrollment increases:

AbhaJindal, director of Kumon tutoring center on Riverside Drive, said more Dublin children have been coming for help in the past couple of years.

"They don’t know how to multiply or divide," she said.

Administrators start making excuses:

George Viebranz, executive director of the Ohio Mathematics and Science Coalition, said it takes time for the benefits of reform math to materialize.

Decades, apparently.

Blame somebody, like the hapless teachers:

"The challenge we face is the teachers, who are key to the success of the program, are products of the system we are trying to change," he said. "There’s a very strong element of professional development that would have to occur over a number of years."

Should have thought of that before implementing the new program.

Curriculum developer tacitly admits it was wrong:

Ken Mayer, a spokesman for the developer of Investigations, said the second edition of the program puts more emphasis on teaching standard algorithms, the traditional ways of solving math problems. That edition has just been released.

And, resorts to spouting cliches:

He said Investigations improves on the traditional "drill and kill" model, because that method leaves students ill equipped for real-world situations.

"A mechanical procedure can be obscure when they have to apply their knowledge to more and more difficult problems," he said. "(Knowing) when they should be adding, multiplying, that’s when they get in trouble."

A mechanical procedure can also be obscure when it has only been superficially taught. And, students being able to "apply their knowledge to more and more difficult problems" invariably requires the application of some procedure, preferably one that has been taught to mastery.

The article alleges that more recent cohorts score better on achievement tests in the lower grades. I'd like to see those tests.

Rooker advises students on college admissions. She said she has asked 85 college admissions officers in the past two years what was the first thing they look for in applicants' transcripts. She said each told her it was "the level of difficulty of the courses taken by a student. It is an automatic assumption that if an able student does not take AP courses when his or her high school offers them, then he or she has chosen not to challenge him or herself."

On that scale, she said, "some New Canaan students appear to be slouches. Why? Because even though NCHS offers AP courses, not many NCHS students take them. Such students appear simply to have chosen not to challenge themselves."

"Yet, I have worked with many students from NCHS who have wanted to take AP courses. They are told, though, that this is not a suitable placement for them. They hear things like: their past performance is not good enough, or they haven't been in honors level classes for the previous grade, or there is only one class and it is full."

Rooker said this is especially true of students with scores between 1250 and 1450 on the old SAT scale of 1600. She said she works with students from several other Connecticut high schools, and sometimes finds students with 1200 on their SATs taking four or five AP courses while New Canaan seniors with 1400s take none. "This is not because they are lazy, but because they were prevented from taking AP classes at NCHS by the system of placement." She said some NCHS staffers say the colleges its students apply to know that the school's non-AP courses are very rigorous, but that argument doesn't work if a New Canaan student applies to a college outside New England that doesn't know much about the school.

"My question is this: in a wonderful school like NCHS where there are a plethora of capable students and capable teachers, why are all able students not offered the opportunity to take AP courses?" Rooker said. "If this means changing the selection criteria, then that should be done. If it means adding more sections of some courses, then that should be done. If it means adding more AP courses, then that should be done. Why are many able students left to appeal and petition for placement in an AP courses, hoping to be admitted, only to find out that they aren't? . . . Our students are caught in the trap of not being allowed to take AP courses by their local high school, and then are being judged by college admissions committees for not taking them."

How does the distribution of truth compare to the distribution of opinion? That is, consider some spectrum of possible answers, like the point difference in a game, or the sea level rise in the next century. On each such spectrum we could get a distribution of (point-estimate) opinions, and in the end a truth. So in each such case we could ask for truth's opinion-rank: what fraction of opinions were less than the truth? For example, if 30% of estimates were below the truth (and 70% above), the opinion-rank of truth was 30%.

If we look at lots of cases in some topic area, we should be able to collect a distribution for truth's opinion-rank, and so answer the interesting question: in this topic area, does the truth tend to be in the middle or the tails of the opinion distribution? That is, if truth usually has an opinion rank between 40% and 60%, then in a sense the middle conformist people are usually right. But if the opinion-rank of truth is usually below 10% or above 90%, then in a sense the extremists are usually right.

My response:

1. As Robin notes, this is ultimately an empirical question which could be answered by collecting a lot of data on forecasts/estimates and true values.

2. However, there is a simple theoretical argument that suggests that truth will be, generally, more extreme than point estimates, that the opinion-rank (as defined above) will have a distribution that is more concentrated at the extremes as compared to a uniform distribution.

I remember this when I was growing up. Every year we got a bit of set theory. Now, everyone thinks of "sets" when the old New Math is brought up. However, it did cover a lot of math and required a lot of mastery. With a rigorous course in algebra in 8th grade, I made it to the calculus (pre AP days) track in high school without ANY outside help. You can't do this nowadays. This doesn't mean that there were no problems back then, but at least there was a proper path.

I've been trying to find out whether any kids in Irvington make it to calculus without outside help.

So far I haven't found anyone. The 3 recent graduates with whom I've been in contact all had parents who knew math extremely well. In two cases the parents were employed in math fields. Those parents were actively engaged in their kids' math education. "Actively involved" as in "helping with homework" and reteaching concepts where necessary.

A parent told me the other day that almost no one takes the BC course - perhaps only 10 to 12 kids per class.

This isn't fact-checked, and it may be wrong. Awhile back a student told us that there were 40 students in AP calculus, split half and half between AB and BC. Total class size was 120.

I don't have the patience to try to figure out whether 30% of an Irvington high school class is comparable to figures for same-SES peers across the country.

I should check private schools, actually. That would give me a rough comparison.

I do know that five percent of all high school students take AP calculus (I'll find the reference); don't know off-hand how that 5% is divided between AB & BC. [correction: make that 8% and rising, I presume]

I'll see if I can track that down.

stats on AP calculus enrollment

I've found a couple of statistics:

Once upon a time, calculus was the first college-level mathematics course taken by mathematically talented students. The students in first-semester calculus were mathematically motivated, generally well prepared, and they were seeing these ideas for the very first time. This is no longer true. Most of our best-prepared mathematics students arrive in college with credit for at least the first semester of calculus, many of them with credit for both semesters. Despite steady growth in majors in science and engineering, enrollment in first-semester calculus has been flat or slightly declining at both two- and four-year undergraduate programs. It is the College Board’s Advanced Placement Calculus Program that has been growing steadily at 7–8% per year (see figure 1).

In 2004 over 225,000 high school students took the AP calculus exam. This number is far larger than the number of students who took mainstream first-semester calculus in all four-year undergraduate programs in the Fall of 2000. By the time of the next CBMS survey in 2005–06, we can expect that more students will take an AP Calculus exam than will take mainstream Calculus I in the Fall of 2005 in all 2-year and 4-year institutions combined.

[snip]

The same pressures that are pushing Calculus I into the high school curriculum are doing the same for Calculus II. Traditionally, it was a very elite group of students who took BC Calculus, covering the entire two-semester college syllabus. That group of students also grew by 6–8% per year until the mid-1990s. Over the period 1995–98, the rate of growth of BC calculus accelerated to 10–11% per year, a rate that has held up since then. In 2004, the number of students taking the BC Calculus exam exceeded 50,000. It will likely exceed 60,000 by 2005–06, the year of the next CBMS survey.

Very small proportions of students complete advanced mathematics or science courses that provide college credit (such as AP/IB courses). The most popular category among these is AP/IB calculus; even there, only 8% of 2000 graduates completed such a course.

Reading and math are the two crucial elementary school subjects required for high school and life beyond, but British Columbia's elementary math curriculum is crippling learning, especially among disadvantaged students.

B.C. has used what is called a "spiral" curriculum since 1987, following a tradition of emulating U.S. educational practice.

A spiral curriculum runs a smorgasbord of math topics by students each year, the idea being that they pick up a little more of each with every pass. In reality, the spin leaves many students and teachers in the dust.

Ideally, the curriculum should cover fewer topics per year in more depth.

Presently, teachers face having Grade 4 classes who still cannot add 567 + 942 nor multiply 7 x 8 because the Grade 1, 2, and 3 teachers were forced to spend so much time on graphing, polygons and circles, estimating quantity and size, geometrical transformations, 2D and 3D geometry and other material not required to make the next step, which is 732 x 34.

And because elementary math fails to provide a solid foundation, many basically capable students simply give up when faced with the shock of high school algebra, which would be the doorway to advanced technical training at all levels. High school math teachers cannot make up Grades 1 to 7 while teaching Grade 8.

Alarm bells about the math curriculum have been ringing in B.C. since the United States, which used spiralling almost exclusively, registered a dismal performance on the Third International Mathematics and Science Study (TIMSS), a test that comparatively evaluated more than 500,000 students from 15,000 schools in 40 countries, first in 1995 and again in 1999 with the same results.

The B.C. ministry of education, to its credit, realized right away in 1995 that the U.S. performance on TIMSS suggested weaknesses in B.C.'s curriculum.

Also aware of some then-emerging data indicating that students in Quebec -- which had retained a sequential curriculum when B.C. went to the spiral -- were outperforming other Canadian students in math, Victoria commissioned researcher Helen Raptis, now a University of Victoria professor, to compare B.C. and Quebec test results and curricula.

In her report, submitted to the ministry in late 2000, Raptis showed that the average B.C. student was more than two years behind the average Quebec student in math by Grade 10, and explored the extent to which curriculum might be responsible.

[Mike] Feinberg: No, KIPP does not have a fifth grade math curriculum; it has a fifth grade math philosophy, it has a fifth grade math scope and sequence but not a curriculum. We realized early on that trying to view the solution as reinventing the wheel and creating a brand new curriculum didn't make a lot of sense. There're a lot of smart people in this country who've already spent a lot of time working on what is good curriculum at first grade, fifth grade and ninth grade. The issue is not that we don't have good curriculum; the issue is that we're not getting the kids to learn it.

Smith: But what's that all about then, getting the kids to learn?

Feinberg: Getting the kids to master the material.

Smith: No, I understand that but what's the key to that? If the curriculum is reasonably good, then what's the key?

Feinberg: Instructional delivery, being very good at teaching in front of the room, very good at using those resources. Being very good at assessing the students and where they are and re-teaching and whatever, doing whatever is necessary to get the kids to really, truly master the material.

You know, talk about curriculum, if I put in front of you a fifth, sixth, seventh, and eighth grade textbook in math and opened up to page 200 and I jumbled them up, and said, “order them from fifth through eighth grade in order,” you'd have a very tough time because they all look the same. That's because, unfortunately, we have this national strategy of “we're not really going to teach to master, we're going to teach to exposure and over lots and lots of years of kids seeing page 200 in the math book, eventually somehow they're going to learn it. We're going to teach them how to reduce fractions in fifth grade, in sixth grade, in seventh grade, in eighth grade, in ninth grade and continue until finally somehow magically they're going to get it.” Instead of thinking, “let's teach the kids how to reduce fractions at a mastery level in fifth grade, maybe spend a little time reviewing it in sixth grade but let's move on to pre-algebra and let's move on to algebra then.” And that's been our take and so it's not that we have a different math curriculum as much as we have a different math strategy and a different math philosophy.

the semiotics of PBS

This passage tells you why the media continues to carry articles on the need for progressive education to replace ineffective traditional practices:

[Feinberg]: The issue is not that we don't have good curriculum; the issue is that we're not getting the kids to learn it.

Smith: But what's that all about then, getting the kids to learn?

Feinberg: Getting the kids to master the material.

Smith: No, I understand that but what's the key to that? If the curriculum is reasonably good, then what's the key?

C. has not been taught Venn diagrams. He's been tested on Venn diagrams, once, but the subject never actually came up in class.

Math Dad got really activated on that one. Goldstar Homework Mom (this is the mom who's blowing me out of the water on homework supervision, reteaching, and tutoring) actually called me up to commiserate: "The reason J. did well is he just happened to ask the tutor about Venn diagrams the week before the test. That's the only reason he could do them."

As I recall, Math Dad had also just happened to teach his son Venn diagrams before the test....and now my friend Kris tells me she is able to guess what's going to be on the test that hasn't been taught in class ------

question

What is my problem?

Why didn't I just so happen to teach my kid Venn diagrams before the test?

Have I mentioned that Ed and I asked the new principal to move Christopher out of accelerated math and into regular-track math for the remainder of the year?

Then move him back to accelerated math next fall?

Ed came up with this plan. That's "Ed" as in not just another pain in the tuchus parent, Ed.

Don't get me wrong.

Ed is a pain in the tuchus.

Ed is also a person who has spent his entire adult life successfully teaching subject matter content to students ranging from young adult GED students in Newark (Ed taught algebra) to Ph.D. candidates at NYU.

Ed, a person holding a Distinguished Teaching Award.

Ed, a guy who knows a thing or two about education.

When Ed came up with this plan I thought: Fantastic plan! It works! It works for everyone! Win-win! YAYYYYY!!!

We'd be out of Ms. K's hair; Ms. K would be out of our hair; Christopher would learn pre-algebra to mastery in his new class and algebra at home; in the fall he would enter a class taught by a teacher who would be getting:

a) a student who knows his stuff

b) a set of parents so grateful to be done serving as Emergency Math Reteachers that teacher & principal could count onnot hearing one word from them all school year

Sounds like an offer you can't refuse, right?

Wrong.

School can move Christopher down. Here in Irvington, that's a lock. No request to move down is ever denied. Quite the opposite, in fact. Requests to move down are encouraged.

So Christopher can move down.

School can't promise to move him back up come fall. Maybe he'll move back up, maybe he won't. School will decide, not us. School won't be consulting with us, either. School is the decider.

That's Irvington.

No promises.

No consultation.

Certainly no guarantees of achievement - no guarantees child will even be allowed to try to raise his achievement.

We've worked long and hard on our goal of having Christopher take algebra in the 8th grade.

Christopher has worked long and hard.

Hell, people here at ktm have worked long and hard. I've taken just about every piece of advice anyone here ever offered me, up to and including instructivist's recent Comment about doing circle graphs using classroom grade distributions.*

The whole family has been committed to this effort. We've invested hundreds of dollars in supplemental workbook and texbook costs, thousands of dollars more in lost work time for me.

School can't promise to help us reach our goal, a goal 80% of 7th graders at KIPP can be reasonably confident they'll be reaching next year.$21,000 per pupil spending; highest property taxes in the country; school is not interested in our goals for our child's education.

Actually, it's worse than that. School is openly indifferent to our goals for our child's education. On occasion school has been openly hostile to our goals.

School can't promise to move him back up.

No reason given.

result: Christopher is staying put.

And I'm teaching Venn diagrams.

back on topic

As advised by our math chair, I am cruising "free worksheets online;" plan to post what I find. If any of you has resources, I'd appeciate your letting me know. Thanks!

*Christopher loved that problem. He insisted on doing a circle graph of what he surmises to be a typical distribution of grades in Ms. K's class. After he did it he said, "Wow. You can really see how many kids aren't learning math very well."